civonamzuk

Interior angles are 30-60-90

If a short leg AB = 10 then the hypotenuse BC = 20   NOT 24!!! 

correct your question

btw, cos of ABM is cos(30º)

tan(R) = 5 * sin(R)

Angle R ≈ 78.46304º

PQ = sin(R) * QR

Circle Gamma is the incircle of triangle ABC and is also the circumcircle of triangle XYZ. Point X is on BC, point Y is on AB, and point Z is on AC. If angle A=40 degrees, angle B=62 degrees, and angle C=78 degrees, what is the measure of angle YXZ?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

∠YXZ = 180 - 1/2[(180 - 62) + (180 - 78)] 

or

∠YXZ = 1/2(62 + 78)

∠APB = arctan(2/1) = 63.43494882º

∠CPQ = 45º

∠APQ = 180 - (∠APB + ∠CPQ) = 71.56505118º

sin(∠APQ) = 0.948683298 

Circle B area = 64pi

Circle A area = (64pi) / 9

Circle A radius = sqrt(64 / 9)

Circle A diameter = 2[sqrt(64 / 9)]

D = 5 ¹/₃ cm

The distance between ω₁ and ω₂ seems to be identical to the length of a radius of a circle ω₂.

The ratio of ab:bc must be 1:3      (The length of ac is irrelevant)

3 : 9 = 8 : x

3x = 72

x = 24

There are a couple of similar right triangles: APC and BQC

AB = 5          AP = 1          AC = x             BQ = 3             BC = 5 - x

AP / AC = BQ / BC

1 / x = 3 / (5 - x)

4x = 5 

x = 1.25

AC = 1.25            BC = 3.75 

PC = sqrt(AC² - AP²) = 0.75

QC = (PC * BQ) / AP         QC = 2.25

PQ = PC + QC    

PQ = 3 cm

(43 + 8)/4 = 12.75  

x + 3x - 9 = 43       x = 13        3x = 39

x + 3x - 5 = 43       x = 12        3x = 36

x + 3x - 8 = 43       x = 12.75          3x = 38.25       Wrong!!!

At a particular school with 43 students, each student takes chemistry, biology, or both. The chemistry class is three times as large as the biology class, and 8() students are taking both classes. How many people are in the chemistry class?

It's NOT 8 !!!

It's either 5 or 9.

(43 + 5)/4 = 12              chemistry class = 3 * 12 = 36

(43 + 9)/4 = 13              chemistry class = 3 * 13 = 39